When Residuals Spread Out
In “Reading a Residual Plot for Model Fit,” you learned that residuals should form a roughly pattern-free band around zero when a straight line describes the relationship reasonably well. In “Curved Residual Pattern Means Nonlinear Relationship,” you saw how an orderly bend can signal that a line systematically overpredicts in some regions and underpredicts in others. A different warning sign is a change in how widely the residuals scatter.
Imagine following the residuals from left to right in a plot. If the points form a narrow band at one end and spread into a wider band at the other, the pattern resembles a fan. The residuals may still be centered roughly around zero, with both positive and negative values, but their sizes are not similar throughout the plot. This changing spread matters because a prediction in a wide part of the fan is surrounded by more variable errors than a prediction in a narrow part.
The key question is not just whether a prediction falls within the observed \(x\)-range, as discussed in “Reliability Within the Range of Data.” Ask also how much residual scatter there is near the \(x\)-value of interest. If nearby residuals vary widely, an individual response can be considerably above or below the fitted value. If nearby residuals stay close to zero, individual responses tend to be more tightly clustered around the line.
A fan shape does not automatically mean that the fitted line’s average trend is curved. Curvature is a pattern in the direction of residuals across \(x\): for example, residuals tend to be positive at both ends and negative in the middle. A fan shape is a pattern in their spread: residuals tend to be more dispersed in one region than another. A plot could show changing spread without a clear curve, or it could show both features at once.
What the Fan Means for Predictions
A fitted line gives a predicted response, \(\hat{y}\), for each \(x\). The residual, \(y-\hat{y}\), measures how far an observed response is from that prediction. As “Using Residual Standard Deviation to Gauge Prediction Error” explains, the residual standard deviation \(s\) gives an overall summary of typical residual size. But one overall summary can hide important local differences. If the fan widens, residuals near one end may often be much larger in magnitude than the residuals near the other end.
That difference affects how confidently you should use a fitted value for a particular case. Where residuals are widely scattered, the line may still describe the general trend, but the response for an individual case is harder to predict precisely. Where residuals are tightly clustered, an individual response is typically closer to the fitted value. This is a comparison of prediction reliability across regions, not a promise that a particular observation will have a small or large residual.
Be specific about which region is less dependable. For a fan that widens as \(x\) increases, predictions at higher \(x\)-values have more variable residual errors than predictions at lower \(x\)-values. For a fan that narrows as \(x\) increases, the comparison goes the other way. If the residual plot is against fitted values, describe the region by the size of the predicted response, rather than automatically calling it “high \(x\).”
Changing spread also cautions against using the overall \(s\) as if it described the typical error equally well everywhere. The value of \(s\) still summarizes the residuals overall, but it does not tell you that the scatter is the same across the full range. Do not turn a visual fan pattern into a precise local error estimate unless the problem supplies an appropriate method and information for doing so.
Worked Examples: Reading Changing Spread
Worked Example: A Fan That Widens as \(x\) Increases
Original AP-style question. In an invented equipment-monitoring exercise, \(x\) is hours of operation and \(y\) is vibration level in millimeters per second. A residual plot from a fitted line shows approximate residuals of \(-1, 1, -1.5, 1.5, -3, 3, -5,\) and \(5\) at increasing operating times. Describe the spread and compare the reliability of predictions at lower and higher operating times.
State. We want to determine whether the line’s prediction errors have similar spread throughout the observed operating-time range.
Plan. Compare the magnitudes of the residuals from early to late operating times. Since residuals are observed vibration minus predicted vibration, positive and negative signs show over- and underprediction; their magnitudes show the vertical distance from the line. This is a descriptive residual-plot assessment, not an inference procedure.
Do. The first two residual magnitudes are \(1\) and \(1\) millimeter per second. The last two are \(5\) and \(5\) millimeters per second. For a broader comparison, the mean absolute residual among the first four displayed points is
The mean absolute residual among the last four is
These summaries are just a way to compare the displayed residual magnitudes; they are not a formal prediction-interval calculation. The residuals remain on both sides of zero, but their vertical spread becomes greater as operating time increases.
Conclude. The plot shows a widening fan, so individual vibration levels are less reliably predicted at higher operating times than at lower operating times. The pattern indicates changing residual spread; by itself, it does not show that the line is systematically high or low at high operating times.
Worked Example: A Fan That Narrows as \(x\) Increases
Original AP-style question. In an invented school transportation exercise, \(x\) is the number of students riding a route and \(y\) is the route’s average morning delay in minutes. A residual plot shows approximate residuals of \(-6, 5, -4, 4, -2, 2, -1,\) and \(1\) minutes as the number of riders increases. Which part of the data has less reliable individual predictions?
State. We will compare the spread of residuals for routes with fewer riders and routes with more riders.
Plan. Group the first four and last four residuals in the displayed order, then compare their mean absolute magnitudes. Absolute values are useful here because a positive residual and a negative residual both contribute to spread.
Do. For the first four residuals, the mean absolute residual is
For the last four residuals, it is
The residuals are much more spread out among the displayed observations at lower rider counts. As rider count increases, the residual magnitudes generally become smaller. This is a narrowing fan, not a widening one.
Conclude. Individual delay predictions are less dependable for routes with fewer riders, where the residuals vary more widely around the fitted line. Predictions for routes with more riders are generally more precise in this data. This comparison concerns the observed residual pattern; it does not guarantee that every lower-rider route will have a larger prediction error.
Worked Example: Spread Changes Across Fitted Values
Original AP-style question. In an invented community energy exercise, a residual plot is displayed against fitted monthly electricity use, measured in kilowatt-hours. For four observations with fitted values near 200 kilowatt-hours, residuals are \(-1, 1, -2,\) and \(2\) kilowatt-hours. For four observations with fitted values near 800 kilowatt-hours, residuals are \(-3, 4, -5,\) and \(4\) kilowatt-hours. Explain which predictions are less dependable and what the plot does not establish.
State. The question is whether residual spread differs for low and high fitted electricity use.
Plan. Compare the average absolute residual in each group. Because the horizontal axis is fitted value, describe the regions by predicted electricity use, not by an \(x\)-value that is not shown on this plot.
Do. Near a fitted value of 200 kilowatt-hours, the mean absolute residual is
Near a fitted value of 800 kilowatt-hours, the mean absolute residual is
The residual magnitudes are larger in the region of higher fitted monthly use. In both groups, residuals include both positive and negative values, so these displayed values do not show a consistent direction of error within either group.
Conclude. Individual monthly-use predictions are less dependable in the higher-fitted-use region because residuals there have greater spread. The plot shows a difference in variability; it does not, on its own, prove that the line is biased upward or downward in that region, identify a cause for the changing spread, or determine the best alternative model.
Common Mistakes and AP Exam Tip
- Confusing a fan with a curve. Curvature concerns a systematic change in residual signs across \(x\). A fan concerns a change in the residuals’ vertical spread. Describe the feature you actually see; if both appear, mention both.
- Reversing which region is less reliable. The wider end of the fan has more variable residual errors and less dependable individual predictions. State the direction of the fan and name the corresponding region.
- Claiming that wide spread means the line always overpredicts or underpredicts. Large positive and negative residuals both indicate spread. A directional claim requires evidence that residuals tend to have one sign in that region.
- Treating one distant point as a fan. One unusually large residual may be an unusual point, as discussed in earlier tutorials. A fan is a broader change in spread across a region, not simply one point far from zero.
- Using the overall \(s\) as a local guarantee. The residual standard deviation summarizes overall residual size; a fan warns that this size may not represent every part of the data equally well.
For full-credit communication, name the changing-spread pattern, identify where the residuals are more dispersed, and connect that region to less reliable individual predictions. For example: “The residual plot fans out as operating time increases. The vibration predictions are less dependable at higher operating times because residuals there have greater vertical spread. The plot does not show a consistent direction of error in that region.”
Check Your Understanding
Use the direction and width of the residual pattern to compare prediction reliability.
- A residual plot has a narrow band at low \(x\) and a wide band at high \(x\). In which region are individual predictions less dependable, and why?
- A fan narrows from left to right. What does that suggest about prediction reliability across the horizontal axis?
- How does a fan-shaped pattern differ from a curved residual pattern?
- Residuals are widely spread around zero at high fitted values, with both positive and negative signs. What can you conclude, and what directional claim should you avoid?
- Why might one overall residual standard deviation fail to describe prediction error equally well throughout a fan-shaped plot?